Annals of Mathematics

The Dimension of the Maximal Measure for a Polynomial MapAuthor(s): Anthony ManningSource: Annals of Mathematics, Second Series, Vol. 119, No. 2 (Mar., 1984), pp. 425-430Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/2007044 .

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Annals of Mathematics, 119 (1984), 425-430

The dimension of the maximal measure for a polynomial map

By ANTHONY MANNING

Let f: C -> C be a polynomial of degree d ? 2. The most interesting dynamics of f takes place on its Julia set J(f), an invariant compact perfect set which can be defined, for example, as the closure of the repelling periodic points. Brolin constructed the equilibrium distribution m on J(f), an ergodic measure which is also the weak limit as n x-> of a probability measure distributed equally over the dn nth inverse images of almost any point. We shall prove the following:

THEOREM. Suppose that each critical point c of a polynomial f: C -> C (i.e. a point such that f'(c) = 0) satisfies fn(c) -/"> x and c e J(f). Then the equilibrium distribution m on J(f) has Hausdorff dimension 1.

Remarks. Recent work in [6], [7], [3], [9] has shown that, even for rational maps, the same weak limit m is the unique measure with the maximal entropy, namely log d. That the topological entropy of our maps f is log d can be seen as follows. Since J(f) does not contain a critical point it is hyperbolic; see [4]. Guckenheimer then exhibits it as a quotient of the shift on d symbols, which must have h(f) < log d. The opposite inequality is in [11].

The condition that critical points do not go to infinity ensures that J(f) is connected and contains a Jordan curve. See Brolin [2] for this and other facts about Julia sets. So the Hausdorff dimension, which is defined in [5], of J(f) satisfies HD( J(f)) > 1 and clearly HD( J(f)) is an upper bound for the Haus- dorff dimension of m. For example, the unit circle, I z = 1, is J(g) for the polynomial g(z) = zd. The hyperbolicity of g ensures that any polynomial f near enough to g has J(f) contain a Jordan curve near J(g). If f has degree d then J(f) is this Jordan curve. If f is given by f(z) = Zd + p for some small p then Ruelle showed in [13] that HD(J(f))= 1 + I p 12/4 log d plus higher order terms. For g the equilibrium distribution m is just the normalised one-dimensional Lebesgue measure around the unit circle. The Hausdorff dimension of a probabil- ity measure t is defined as HD(t) = inf{HD(Y); 1i(Y) = 1); see [14]. For g we clearly have HD(m) = 1. The surprise is that this property holds also for the

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426 ANTHONY MANNING

perturbations f. So long as f]J(f) is topologically conjugate to glJ(g) the conjugacy, which carries the Lebesgue measure on the unit circle to m on J(f) and increases HD(J), avoids changing HD(m). The theorem was first suggested by the author's attempts to estimate HD(m) by computer for small p. Further calculations suggest that the theorem does not extend to rational maps f. In Section 15 of [2] Brolin shows that J(f) has capacity 1 in the sense of potential theory. This does not seem to be related to our result since it only holds for monic polynomials and does not require the Julia set to be connected.

We shall need the following lemma whose proof we postpone.

LEMMA. Let B be an open subset of C, f: B -> C an analytic map and J C B a compact invariant set. Suppose that f' is never zero on J and let [ be an f-invariant ergodic probability measure on J. Then the Hausdorff dimension of A, its entropy and its Lyaputnov exponent X,, = A(log If I) are related by

HD([t) = h(f)/XA.

Proof of theorem. Since f is analytic it has only one exponent X,, for each a, not two. See [12] for the definition of these exponents. Since hm(f ) = log d it suffices to prove that Xm = log d. By Brolin's construction [2], m is the weak limit of d fy=Wvy for any given w with at most two exceptions. Thus

Xm=rn L lognf'(y)l = lim '1 log H If'(y)i. n o+ CCfdy=w no -, cY=W

Now f' is a polynomial of degree d - 1 with roots cl,... ,cd- say (not necessarily distinct), the critical points of f. Then f'(z) = dadni(z - ci) where adZd is the leading term of f(z) and so

d-1X

Xm = lim d-nlog((dad)d nH H IY -cI. n -, oo i=1 f'y=w

Consider the equation fn(y) = w for y and the corresponding equation fn( u + c,) - w = 0 for u = y - ci. This latter is a polynomial equation of degree dn whose leading coefficient is a(dn -1)A(d- 1) The constant term in the equation is fn(ci) - w as can be seen by putting u = 0. Thus

f Iy _ Cil = IHU (- 1)d(fn(Ci) - w)la(d n-1)1(d-1) fny =W

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DIMENSION OF THE MAXIMAL MEASURE 427

Hence, we have

Xm = lim d~ - (log If"(cj) - WI - (d n - 1)7 (d - 1)log ad)

+ d nlog( dad)}

d-1 = log d + 1: lim d fnloglffl(Ci) - WI + lim d -nlog ad

= log d,

using the hypothesis that each ci is not in the basin of attraction of o. E

Remark. If some critical point is attracted to infinity then the last equation shows that Xm ? log d and, provided there is no critical point in the Julia set, the lemma gives HD(m) ? 1.

Proof of lemma. The proof will resemble that in [10] but first we use the analytic property of f to estimate its stretching in two dimensions.

Given a small positive a choose a compact neighbourhood K of I on which If a. Here we use the hypothesis that no critical point belongs to J. Choose a

finite open coverV sof K by open balls on each of which f' varies by at most a 2 and let L be a Lebesgue number forV/. If x, y E- A E- V~ let -y: [0, 1] ->C be the straight path from x to y. Then

fAy) - fAX) I f(f-r)'(t) dt =f f(-y(t)) -y'(t) dt = I y - x I ff'(y(t)) dt

so that

If(Y) - fAO) - If'(x)I < la 2

since f' varies by at most a 2 along -y. Hence

(1) PxIf'(x) I < If(y) - f(x) I/Iy - xlI < (1 + xa) If'(x)I1 and so

(* e-a If (x)I < If(y) - f(x)lI/Iy - xlI < ea If (x)I. Consider fJ] and recall Bowen's approach in [1] to h,,(f). For U c I define

n = r,( U) as the largest non-negative integer such that fi U is contained in some set of W1for 0 ? i < n. Then for Y c J define

m,,(Y, A) = lim inf{f ure,,exp - Xry(U)}

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428 ANTHONY MANNING

where the infimum is taken over countable covers O& of Y for which each summand exp - Arn(U) is less than e. Then h'w(f, Y) is defined as sup{ A; m,(Y, A) = x or equivalently as inf{ A; m_(Y, A) = 0). Bowen proved in [1] that the measure-theoretic entropy is inf{ h(f, Y); P(Y) = 1) and also h(f, G,,) where G, denotes the generic points of p. Here h(f, Y) = suph4(f, Y).

For our given a put n-1

GIr= {x E G,,; nX, - E loglf'(fix)l < na for all n > r i=O

and notice that G, = UrG, r -For a set U C J take x c U n Gtr and suppose n = nr(U) > r. Then

mesh ?> diam(f-'U) ? sup lfn- ly - f'n- xI

> supYEUy - xI If (x)I If'(fn2x)I

xexp - (n - I)a by(*) n-2

2 I diam U.exp E log If'(fix)I exp - (n - 1) a i=O

22diamU.exp(n - 1)(X, - 2a)

if we use x E Gt r -Also

L < diam(fnU) < 2 supy FUly -fnXI

< 2 diam U. If'(x)I ... If'(fn- 1x)I exp n a n-1

-2 diam U.exp na + E logIf (fixI)I i =o

< 2diamU.expn(X, + 2a).

Thus

2Lexp - n(x, + 2a) < diamU < [2meshv.exp(x, - 2a)]exp - n(x, - 2a)

This inequality relating diamU and nX, enables us to relate HD(O) and h(f). If , = HD(Y) for Y c GAC J with p(Y) = 1, put Yr = Y n G .A r Take a cover 9r of Yr by subsets of Y, fine enough that nr(U) > r for U E 1&r' such that

e (diamU)+ < 2 U E= tr,

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DIMENSION OF THE MAXIMAL MEASURE 429

Then Y = U0?=Y,

and it is covered by 9/= UW IT, Now

m,(Y, (13 + a)(X, + 2a)) < E exp - nr(U)(f3 + a)(X, + 2a) Useq

< (2/L) + a E, (diam.U) + a

usei

< (2/L)?+ajE2-r = (2/L)l+a r

so that h(f,Y) < (/3 + a)(X, + 2a) and h] < Bx.,,. On the other hand mw(Yr, hi + a) = 0 for each r so we can choose a cover

fof Yrwith nr(V) > r for each V and

E exp - n_(V)(h, + a) < 1. vet

Now

E (diam V)(h+,a)AxA-2a) Vet

< [2meshV.exp(X, - 2a)]( xa)px-2a) e - (V)(

< [2 mesh -a. exp (X Z-2 a) (h,+a)/ X2 -a)

Thus HD(Yr) < (h,, + a)/(Xl - 2a). This holds for each r and a is arbitrary; so /3 = HD(Y) < h./Xwhich completes the proof of the lemma. n

Remark. Manei told the author that an argument like that in his paper [8] could extend our theorem to the case where f does have critical points in its Julia set.

UNIVERSITY OF WARWICK, COVENTRY, ENGLAND

REFERENCES

[1] R. BOWEN, Topological entropy for non-compact sets, Trans. A.M.S. 184 (1973), 125-136. [2] H. BROLIN, Invariant sets under iteration of rational functions, Arkiv for Mat. 6 (1965),

103-144. [3] A. FREIRE, A. LOPES and R. MANE, An invariant measure for rational maps, Bol. Soc. Bras.

Mat. 14 (1983), 45-62. [4] J. GUCKENHEIMER, Endomorphisms of the Riemann sphere, Proc. Symp. Pure Math. 14

(1970), 95-123. [5] W. HunEwIcz and H. WALLMAN, Dimension Theory, Princeton University Press, Princeton,

1941.

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430 ANTHONY MANNING

[6] M. Yu. LYUBICH, Entropy of analytic endomorphisms of the Riemann sphere, Func. Anal. and its Appl. 15 (1981), 300-302.

[7] , The maximum-entropy measure of a rational endomorphism of the Riemann sphere, Func. Anal. and its Appl. 16 (1982), 309-310.

[8] R. MANE, A proof of Pesin's formula, Ergod. Th. & Dynam. Sys. 1 (1981), 95-102. [9] , On the uniqueness of the maximising measure for rational maps, Bol. Soc. Bras. Mat.

14 (1983), 27-43. [10] A. MANNING, A relation between Lyapunov exponents, Hausdorff dimension and entropy,

Ergod. Th. & Dyn. Syst. 1 (1981), 451-459. [11] M. MisiumRwicz and F. PRZYrYCKI, Topological entropy and degree of smooth mappings,

Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 25 (1977), 573-574. [12] YA. B. PESIN, Characteristic Lyapunov exponents and smooth ergodic theory, Russ. Math.

Surveys, 32 (1977), no. 4, 55-114. [13] D. RUELLE, Repellers for real analytic maps, Ergod. Th. & Dyn. Syst. 2 (1982), 99-107. [14] L -S. YOUNG, Dimension, entropy and Lyapunov exponents, Ergod. Th. & Dyn. Syst. 2

(1982), 109-124.

(Received July 29, 1983)

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